Hi, my name is Anthony Rhodes
and this is Complexity Explorer's tutorial
on Vector and Matrix Algebra.
so this... unit is broken up into three subsections total
in the first subsection we'll be looking at...
domains of numbers, set theory and vectors.
in the second section...
we'll be exploring the notion of matrix
which is really the heart of this unit.
And some of the predominant
algebraic and arithmetic properties of matrices.
At last, in the third section
we will be looking at some applications of matrices At last, in the third section
we will be looking at some applications of matrices
with an eye towards... we will be looking at some applications of matrices
with an eye towards...
replications in complex networks and complexity
read large, in particular.
One of the applications we look at, for instance is...
a way to efficiently compute
the power of a matrix,
that's a technique known as diagonalization.
we'll also, in addition to that,
look at some geometric
properties and applications of matrices
known as linear transformations and rotations
And lastly, we will be looking at
applications that relate matrices and matrix computations
to probabilistic models
And that's a technique, or topic, known as Markov chains
So before we begin,
i'd like to first help motivate the study of matrices
by just mentioning briefly a few examples
of places where we can use matrices
to study interesting phenomena
and applied math in science
so, for instance, one of the all-time
classic applied problems really in the history of
of math and science
has to do with curve fitting
what i mean by that is if I perform an experiment has to do with curve fitting
what i mean by that is if I perform an experiment
and collect data
I would like to
for the purposes of prediction in inference
to build a model that best fits that data
In a general setting, this is known as
the problem of regression.
So as it turns out
in learning this best-fit curve
i can include the relationship between the data
and the curve I'd like to build,
or the model i'd like to build
using matrices
And those equations are very well known and well studied
And they're known as 'the normal equation'.
You see oftentimes matrices
as representing operations
or quantum computations
and... statistical physics, for instance
a matrix can include relationships
in a vector field
if I want to know how particles in room
interact, i can do that.
actually, as it turns out, via matrix multiplications...
And, of course, network theory and graph theory
replete with examples that apply
ideas related to matrices.
and let's cite just a few of those
so, for instance,
if I'm in city A
and I'd like to find the most efficient route to city B
and we can think of this as, maybe, the code to Google Map problem
which we'll explore later and we can think of this as, maybe, the code to Google Map problem
which we'll explore later
I can query Google Maps or some similar algorithm
get a pretty decent result as to the most efficient
way or efficient route from
city A to city B.
In general, that algorithm, or variant of it is
known as Dijkstra's algorithm
As another example we can think of
in fact, the Internet itself
as representing a complex network or graph
where the nodes, or vertices, in the graph
are computers or servers that connect to
via some link.
So we might wonder when studying a complex network
like the Internet, are some of the nodes
more important, or some of the servers more important, than others.
And we might wonder such a thing because, well
well, if we lost some of those servers,
is it possible that we have catastrophic
failure of the Internet, or some such network
Because we don't want that, so we'd like to take the necessary precautions.